Arrow’s theorem states, colloquially, that there doesn’t exist a voting system that is “fair” in a certain sense. He proved this by showing that there exist cycles of preferences for any group decision procedure with three or more voters, and with at least three choices. An interesting question is, what is the likelihood that individuals preferences will line up in such a way that a cycle exists. In other words, what are the odds that a given election is “fair”, due to the fact that enough people agree. If there are three people and three different choices to make, then there are six possible preference rankings for each person giving a total of 6^3 or 218 different preference orderings. It can be shown that there are 12 orderings for which a cycle develops. So in the case of three persons and three choices, if all choices are equally probably, then the odds that a decision will be “fair” is about 94% (1 – 12 / 218).

The odds of a cycle developing for a given number of voters, with a certain number of choices (assuming equiprobable ranking) was calculated by Niemi and Weisberg. The calculations of the odds of a cycle for three and infinite choices are given below.

Odds of a cycle | ||
---|---|---|

Three Choices | Infinite Choices | |

N=3 | 0.056 | 0.080 |

N=4 | 0.111 | 0.176 |

N=5 | 0.160 | 0.251 |

N=6 | 0.202 | 0.315 |

N=7 | 0.239 | 0.369 |

N=inf | 1.000 | 1.000 |

Another way to think about it is that the odds of a cycle is the probability that a particular vote can be gamed. As an example, let’s assume that there are three friends, Andy (A), Bill (B), and Carrie (C), and they each go out to the movies each weekend. The friends discuss what their order of preference for various movie genres are on the drive to the on the way to the theater, and then vote on which genre will be chosen. Since Carrie is an election expert and the other two don’t know any better they decide that she can decide the form of the election each week, so long as it obeys certain criteria (non-dictatorial, etc). At the theater there are three types of movies to be seen, action movies (a), black comedies (b) and chick flicks (c). The tastes of each of the three friends are random from week to week. Around 5.6% of the time Carrie will able to choose an election system that will allow her to see the genre of her choice, even though a different choice could have allowed either of the other genres to be chosen. If another friend, Damian (D), wanted to join the club, then there is an 11% chance that Carrie could game the vote. In this case she couldn’t guarantee her top choice, but she can still decide among three options.

In my view it is probably always best to look at the choices as the options approaches infinite, since there are alway more options to pick (get popcorn, get lowfat popcorn, don’t get popcorn, sit in the front, sit in the back, sit in the middle, etc. etc). In this case, even if there are seven people voting, there is a 37% chance that the vote might be considered “fair”. It would be interesting to find an asymptotic expansion to see where the 50% tipping point is. In any case, as the number of people voting expands, the odds of a cycle increases to the point where there is no longer likely to be a discernible “will” of the voters.

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